i came across a question today about the distannce between two parallel lines in 2D and i was wondering if i am asked the distance between two parallel lines, could i find the distance between the two parallel planes that these vectors were on in 3D (if the z component was equal to 0 for both planes)?

There are well-known formulae for the distance between a point and a line and between a point and a plane. Finding distance between a point on one line to the other line gives the distance between the two parallel lines. The same basic idea works for finding the distance between two parallel planes.,

i came across a question today about the distannce between two parallel lines in 2D and i was wondering

if i am asked the distance between two parallel lines, could i find the distance between the two parallel planes that these vectors were on in 3D (if the z component was equal to 0 for both planes)?

Two parallel lines are on an infinite number of parallel planes- however, they all have the same distance between them.

Personally, what I would do is choose an arbitrary point on one of the lines, construct the plane perpendicular to the line through that point (very simple since you are given the line and so a vector perpendicular to the plane), determine where the other line intersected the line, then find the distance between those two points.

well if i had for example two lines 3x+7y=-4 and 3x+7y=10 and treated them as parallel planes, with equation x.n=-4 and x.n=10 can i then use the distance between these two planes, e.g. |-4-10|/|n| where n = <3,4>

ok so that is equal to 14/((58)^(1/2)) but if i use x.n=-4 and x.n=10 where n = <3,4,0> then i get |-4-10|/((9+49)^(1/2)) which is equal to the same thing,
so i was wondering if treated parallel lines as planes parallel to the z-plane if i can use this to find the distance?

Suppose that and is a vector.
Then is a plane with normal containing the point .
For any point its distance to the plane is
Just do in general. Donít worry where the z-plane is.